01 - Whisk together flour, sugar, yeast, salt, cinnamon, and nutmeg in a large bowl until thoroughly combined.
02 - In a separate bowl, whisk warm milk, melted butter, and egg until blended. Pour into dry ingredients and mix until soft dough forms.
03 - Turn dough onto lightly floured surface and knead for 5 minutes until smooth and elastic. Add small amounts of flour if sticky.
04 - Place dough in lightly greased bowl, cover with plastic wrap or towel, and let rise in warm place for 1 hour until doubled in size.
05 - Preheat oven to 350°F and line baking sheet with parchment paper.
06 - Mix brown sugar and cinnamon in small bowl until uniform.
07 - Roll dough to 1/2-inch thickness. Cut 12 donuts using 3-inch round cutter with 1-inch center cutter, rerolling scraps as needed.
08 - Transfer donuts to prepared baking sheet. Brush tops with softened butter and sprinkle with cinnamon-sugar mixture, pressing gently to adhere.
09 - Bake for 12-15 minutes until lightly golden and set. Cool on wire rack for 5 minutes.
10 - Whisk powdered sugar, 2 tbsp milk, and vanilla until smooth. Add additional milk if needed for thick, pourable consistency.
11 - Dip tops of warm donuts into glaze, then immediately sprinkle with purple, green, and gold sugars in alternating sections. Let set for 10 minutes before serving.